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Questions tagged [philosophy-of-mathematics]

Philosophy of mathematics asks questions about mathematical theories and practices. It can include questions about the nature or reality of numbers, the ground and limits of formal systems and the nature of the different mathematical disciplines.

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Probability calculus and Quantum Mechanics

I am not an expert and probably this question highlights this. Anyway, is the probability calculus used in Quantum Mechanics? Does the concept of probability adopted in Quantum Mechanics satisfy the ...
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1answer
66 views

Why is ZFC not as susceptible to Gödel's incompleteness as was the Principia Mathematica?

So, from what little I have read (such as this answer), it appears to be that one reason why the program of Logicism, as laid out in the Principia Mathematica, failed was that its goals (of finding a ...
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Discovery VS Invention in Mathematics [duplicate]

Asking specifically in the context of philosophy of Mathematics, on what basis do we classify or should classify a new expression as a Creation of Mind (Invention) OR a Discovery?
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Philosophy - Does Einstein's Block Universe theory prove Nietzsche's Eternal Return theory is true?

If the Past, Present, and Future all exist in exactly the same way, then every single moment would be a ‘Now’ moment for me. it would also mean that me being dead in the future is equally real in the ...
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Philosophy of Logic – Converting formal proofs to the sound deductive logical inference model [on hold]

How do we convert the formal proofs to theorem consequences of symbolic logic to conform to the sound deductive logical inference model? What would be the benefits of doing this? PROPOSED ANSWER: ...
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1answer
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Could generalization of scientific theories be possible by just adding an ad hoc hypothesis?

In a seventeenth century world the Newtonian model did mostly very well to describe how gravity works in the universe and did well with most empirical evidence of that time. Of course now we know that ...
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9answers
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Is a proof still valid if only the author understands it?

Some time ago I was reading about the recent Shinichi Mochizuki's proof for the famous ABC conjecture. It's enormous and so incredibly difficult that at that time virtually nobody was able to ...
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2answers
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For a mathematical realist, is there a distinction between real mathematical objects and constructed mathematical objects?

Mathematical realists believe that mathematical entities exit independently of human minds. Mathematical objects have an objective independent existence, and they are discovered by mathematicians, not ...
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25answers
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Why is there something instead of nothing?

A simple but fundamental question. The "something" means the whole Universe (known and unknown), it could be represented as the reality version of the set of all sets, which is itself debated. It ...
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1answer
312 views

Was Kant an Intuitionist about mathematical objects?

In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-...
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3answers
652 views

What are the discoveries that have been possible with the rejection of positivism?

I am wondering if the rejection of the positivism movement in philosophy lead to any major discoveries in mathematics and natural sciences? I am thinking it might have been able to contribute to those ...
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4answers
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What does this Jacques Hadamard quote mean?

What does this Jacques Hadamard quote mean? The shortest path between two truths in the real domain passes through the complex domain. Is this a philosophical statement? what is its mathematical ...
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1answer
235 views

Is there any physics-model version of Tegmark's hypothesis?

Tegmark's mathematical universe hypothesis is very interesting (https://en.m.wikipedia.org/wiki/Mathematical_universe_hypothesis) but it has virtually no support among physicists because it is too ...
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The nature of Nominalist Formalism

In this entry in the stanford encyclopedia of philosophy, it is stated that the theory of nominalist formalism deals with the metatheory problem of formalism as follows: Commendably, Goodman and ...
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2answers
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can we reason about logic?

People who study mathematical logic make arguments about logic itself. So it seems that people take for granted an "intuitive logic" (otherwise, how would they form arguments?). So the observation is ...
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3answers
147 views

Mathematical Consensus

Can anyone give me a reason why mathematics may require consensus to determine the quality of knowledge from the general mathematical community? Also what would be the counter to such a claim, as in ...
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1answer
207 views

Are mathematical results influenced by the way we reason?

Intuitions of mathematicians, and the mathematics they develop, are ostensibly influenced by whether they primarily rely on visual_spatial and/or verbal_symbolic reasoning skills. Is it fair to say ...
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2answers
168 views

Where can I learn about the philosophy behind mathematical and logical proofs?

I'm looking for something that dives into the philosophical idea of a "proof," and explains how the subjects of mathematics and logic deal with it. Does anyone have any book or article recommendations ...
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2answers
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Does philosophy of mathematics affect mathematical research?

I am interested in a special case of the general question about whether the philosophy of X has an effect on the research or practice of X. My special interest is in the area of mathematics. I am a ...
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1answer
165 views

Where to start with the philosophy of mathematics?

This may be a duplicate question. What is the best way to get started with the philosophy of mathematics? Given that I know (from university) the basics that are discussed (Set theory, Russell's ...
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1answer
120 views

Is mathematics something real or just an abstraction we created?

Is mathematics something real like something so correlated in our universe which would be different in other theoretically universes or is it just an abstract universe-independent layer/framework we ...
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2answers
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Must the physical phenomenon of the universe be differentiable?

The use of Calculus for the analysis of real-world phenomenon depends entirely on our universe not only being continuous, but being differentiable. By "real-world phenomenon" I mean things like the ...
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1answer
112 views

Can you imagine a completely different logical/mathematical system than that we have?

Can you imagine a different logic and mathematics? For example, with a different arithmetic, or even a universe with no logic or mathematics and contradictions? A non consistent system?...
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Falsification in Math vs Science

In the beginning it was thought that the statement 1+1=0 is false, and necessarily so. However, with the birth of modular arithmetic, it was found that indeed, 1+1 does indeed equal to 0 (in the mod 2 ...
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1answer
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Is there a natural example of a non-self-referential semantic paradox in philosophy?

A commonly studied paradox is the liar's paradox. The liar's paradox is to determine whether "this statement is false". The usual resolution is to state this the sentence is not actually a statement ...
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What was Wittgenstein's argument against Cantor's transfinite numbers and where did he make his objection?

G. E. M. Anscombe had this to say about propositions in Wittgenstein's Tractatus: (page 137) It seems likely enough, indeed, that Wittgenstein objected to Cantor's result even at this date, and ...
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1answer
190 views

Gödel's Results and Philosophy of Mathematics [closed]

Gödel's results essentially conclude that there are True but Unprovable statements in arithmetic. My thoughts are as follows: Axioms form the foundation of mathematics -because we need to assume ...
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2answers
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What does Wittgenstein mean when he says “there are no numbers in logic”?

From the Tractatus: 5.453 All numbers in logic stand in need of justification. Or rather, it must become evident that there are no numbers in logic. There are no pre-eminent numbers. What does ...
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1answer
194 views

Are axioms in mathematics comparable to hypotheses in experimental sciences?

Remark: my question deals more particularly with the axioms of set theory, arithmetic, probability theory, etc. I think the status of the axioms in geometry is clearer. The French fictitious ...
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2answers
430 views

What theorems are most important for the foundation of mathematics?

What are the mathematical theorems which are considered as the most important for the mathematics themselves? By importance I mean foundational to mathematics as a whole or foundational to a good ...
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4answers
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Believing in the axiom of Power Set

I am struggling to find a philosophical reason for believing in the axiom of power set, and I was hoping you can give me some justifications. I am not looking for answers of the form "it's convenient ...
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1answer
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Can pure randomness be computed?

Algorithm for randomness usually use seed, and thus having an unique input it cannot be said to be completely random, so can pure randomness be theoretically computed?
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Are the “laws” of deductive logic empirically verifiable?

"Is Logic Empirical?" strongly suggests a question that I would like very much to get a handle on. That phrase is a title of an article by Hilary Putnam, and, according to synopses/reviews, the ...
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4answers
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Is math arbitrary?

Math at its core begins with calling something true or false and following logic. WE for example call an odd number 2n+1, but what if we called an odd number 2n and flipped it for it to become an even ...
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What are some mathematical fields that can be useful to philosophers?

I am wondering if there's any field in mathematics that can help philosophers define things or help a philosopher make an argument for something. I am just wondering if there's any mathematics that ...
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What is the difference between depth and surface information?

I was looking for an answer to this question: Was Euclid's method of proof axiomatic? While doing so I ran across an abstract of Jaakko Hintikka for an article "What is the axiomatic method?" ...
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Do picture proofs of the Pythagorean theorem make it empirical?

As I understand it, the Pythagorean Theorem, which defines the metric for Euclidean space, is said to be strictly mathematical in the sense that it is derived from a set of purely theoretical axioms (...
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258 views

Was Euclid's method of proof axiomatic?

Euclid's method of proof has often been described in textbooks as axiomatic, but was it really so? And if not, how else can Euclid's method be characterized?
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What is the connection between conscious mind and Gödel's incompleteness in a mathematical universe?

Assume that our universe is a mathematical one, similar to the one that Tegmark proposed (see here). In contrast to what I read there, let's assume that the axioms upon we build the universe are such ...
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Do transfinite sets have practical applications?

This may not qualify as a philosophy question exactly, but I would argue that potential applications of pure mathematics are in the bounds of philosophical interest. Many innovations in pure ...
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101 views

Can math be done without syllogisms? [closed]

Question seems self explanatory. Is there anything in mathematics that can be stated to be true without using a logical syllogism? Had a discussion with somebody about this recently. Sorry if this is ...
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1answer
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What is the relevance of applicability to the natural sciences in pure mathematics?

I think I am coming to a good, new understanding of the relationship of pure mathematics to the natural sciences. A major concern of mine is just how reliable is rigorous (characteristically "pure") ...
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Did any philosopher make the claim that mathematics can be as illusory as visual information?

The Greeks postulated that the world we observe may be just an illusion and Kant based some of his philosophy on that very idea. From that idea, came the idea that mathematical truths are more certain ...
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166 views

What are some philosophical arguments that explain why mathematics allow us to reach a greater truth than empirical evidences?

Is it really the case? Was there a proof of sort that shows mathematical facts are more certain than empirical facts? What are the arguments for and against that claim?
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1answer
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The mereological account of sets

So it has come to my attention that David Lewis, David M. Armstrong and others tried a mereological account of sets. James Franklin states it as: Armstrong adopts David Lewis’s proposal that a ...
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What actually are meaningless symbols?

Some days ago our professor during the course of his lecture wrote the following definition of a polynomial. We say that an expression of the form a0 + a1x + a2x2 + ... + anxn is a polynomial of ...
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How should one interpret modern mathematics if one doesn't believe in infinity?

I am an ultrafinitist. http://en.wikipedia.org/wiki/Ultrafinitism I don't believe there is a such thing as infinity. To me, it is obvious that there has to be a largest number; I just don't know what ...
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3answers
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How much platonism do I need to handle the halting property?

I always considered myself as platonist (in contrast to formalist / finitist) but recently I realized (if this is actually true) that you need a bit of platonism to even make sense of questions like '...
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Number, Category and Set

Can it be said that a number is a category is a set? There is such a variety of ideas on numbers, categories and sets that probably anything one says about them will be controversial, but I was ...